Some things in the universe are pretty much constant. Except when caught in a black hole or penetrating some very dense material, light travels at the same speed pretty much no matter what. But that's simply not the case with time.

The faster you go, the slower time passes for you. Were you to be able to catch a ride on a ship the got up to some ridiculous percentage of the speed of light, you could whip out to the nearest star and back and likely find your grandkids of today would be the age of your grandparents when you returned. Einstein worked all that out - and experiments with high flying aircraft and such have demonstrated that it really does seem to work out that way.

But there's a far easier way to demonstrate that time moves far differently for different people in different situations. And you already are familiar with this phenomenon.